p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.26C23, C4.342+ 1+4, C4.152- 1+4, C8⋊7D4⋊8C2, C4⋊D8⋊32C2, C8⋊2D4⋊18C2, C4⋊C4.141D4, D4.Q8⋊32C2, C2.35(D4○D8), C4⋊C8.87C22, C22⋊C4.33D4, C23.94(C2×D4), C4⋊C4.198C23, (C2×C8).175C23, (C2×C4).457C24, (C2×D8).77C22, C2.D8.50C22, C4.Q8.50C22, (C4×D4).136C22, (C2×D4).198C23, C4⋊1D4.72C22, C4⋊D4.52C22, D4⋊C4.62C22, (C22×C8).156C22, C22.717(C22×D4), C42.C2.32C22, (C22×C4).1112C23, C42.6C22⋊14C2, C22.34C24⋊8C2, (C2×M4(2)).95C22, C42⋊C2.175C22, C2.76(C22.31C24), (C2×C4).581(C2×D4), SmallGroup(128,1991)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.26C23
G = < a,b,c,d,e | a4=b4=c2=e2=1, d2=b2, ab=ba, cac=a-1, dad-1=ab2, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=a2c, ece=bc, ede=a2d >
Subgroups: 428 in 185 conjugacy classes, 84 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, C22×C4, C22×C4, C2×D4, C2×D4, D4⋊C4, C4⋊C8, C4.Q8, C2.D8, C42⋊C2, C4×D4, C4⋊D4, C4⋊D4, C22.D4, C42.C2, C4⋊1D4, C22×C8, C2×M4(2), C2×D8, C42.6C22, C4⋊D8, C8⋊7D4, C8⋊2D4, D4.Q8, C22.34C24, C42.26C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, 2- 1+4, C22.31C24, D4○D8, C42.26C23
Character table of C42.26C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 8 | 8 | 8 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ9 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ10 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ11 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ12 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ13 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ14 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ15 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ16 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ17 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 0 | 2√2 | 0 | 0 | 0 | orthogonal lifted from D4○D8 |
| ρ22 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| ρ23 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | 0 | -2√2 | 0 | 0 | orthogonal lifted from D4○D8 |
| ρ24 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 0 | 2√2 | 0 | 0 | orthogonal lifted from D4○D8 |
| ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | 0 | -2√2 | 0 | 0 | 0 | orthogonal lifted from D4○D8 |
| ρ26 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from 2- 1+4, Schur index 2 |
►On 64 points
Generators in S64
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)(33 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)(49 50 51 52)(53 54 55 56)(57 58 59 60)(61 62 63 64)
(1 23 25 17)(2 24 26 18)(3 21 27 19)(4 22 28 20)(5 15 63 10)(6 16 64 11)(7 13 61 12)(8 14 62 9)(29 36 41 38)(30 33 42 39)(31 34 43 40)(32 35 44 37)(45 55 58 50)(46 56 59 51)(47 53 60 52)(48 54 57 49)
(1 4)(2 3)(5 9)(6 12)(7 11)(8 10)(13 64)(14 63)(15 62)(16 61)(17 22)(18 21)(19 24)(20 23)(25 28)(26 27)(29 35)(30 34)(31 33)(32 36)(37 41)(38 44)(39 43)(40 42)(45 57)(46 60)(47 59)(48 58)(49 50)(51 52)(53 56)(54 55)
(1 48 25 57)(2 58 26 45)(3 46 27 59)(4 60 28 47)(5 44 63 32)(6 29 64 41)(7 42 61 30)(8 31 62 43)(9 40 14 34)(10 35 15 37)(11 38 16 36)(12 33 13 39)(17 49 23 54)(18 55 24 50)(19 51 21 56)(20 53 22 52)
(1 29)(2 30)(3 31)(4 32)(5 58)(6 59)(7 60)(8 57)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 36)(18 33)(19 34)(20 35)(21 40)(22 37)(23 38)(24 39)(25 41)(26 42)(27 43)(28 44)(45 63)(46 64)(47 61)(48 62)
G:=sub<Sym(64)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,23,25,17)(2,24,26,18)(3,21,27,19)(4,22,28,20)(5,15,63,10)(6,16,64,11)(7,13,61,12)(8,14,62,9)(29,36,41,38)(30,33,42,39)(31,34,43,40)(32,35,44,37)(45,55,58,50)(46,56,59,51)(47,53,60,52)(48,54,57,49), (1,4)(2,3)(5,9)(6,12)(7,11)(8,10)(13,64)(14,63)(15,62)(16,61)(17,22)(18,21)(19,24)(20,23)(25,28)(26,27)(29,35)(30,34)(31,33)(32,36)(37,41)(38,44)(39,43)(40,42)(45,57)(46,60)(47,59)(48,58)(49,50)(51,52)(53,56)(54,55), (1,48,25,57)(2,58,26,45)(3,46,27,59)(4,60,28,47)(5,44,63,32)(6,29,64,41)(7,42,61,30)(8,31,62,43)(9,40,14,34)(10,35,15,37)(11,38,16,36)(12,33,13,39)(17,49,23,54)(18,55,24,50)(19,51,21,56)(20,53,22,52), (1,29)(2,30)(3,31)(4,32)(5,58)(6,59)(7,60)(8,57)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,36)(18,33)(19,34)(20,35)(21,40)(22,37)(23,38)(24,39)(25,41)(26,42)(27,43)(28,44)(45,63)(46,64)(47,61)(48,62)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,23,25,17)(2,24,26,18)(3,21,27,19)(4,22,28,20)(5,15,63,10)(6,16,64,11)(7,13,61,12)(8,14,62,9)(29,36,41,38)(30,33,42,39)(31,34,43,40)(32,35,44,37)(45,55,58,50)(46,56,59,51)(47,53,60,52)(48,54,57,49), (1,4)(2,3)(5,9)(6,12)(7,11)(8,10)(13,64)(14,63)(15,62)(16,61)(17,22)(18,21)(19,24)(20,23)(25,28)(26,27)(29,35)(30,34)(31,33)(32,36)(37,41)(38,44)(39,43)(40,42)(45,57)(46,60)(47,59)(48,58)(49,50)(51,52)(53,56)(54,55), (1,48,25,57)(2,58,26,45)(3,46,27,59)(4,60,28,47)(5,44,63,32)(6,29,64,41)(7,42,61,30)(8,31,62,43)(9,40,14,34)(10,35,15,37)(11,38,16,36)(12,33,13,39)(17,49,23,54)(18,55,24,50)(19,51,21,56)(20,53,22,52), (1,29)(2,30)(3,31)(4,32)(5,58)(6,59)(7,60)(8,57)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,36)(18,33)(19,34)(20,35)(21,40)(22,37)(23,38)(24,39)(25,41)(26,42)(27,43)(28,44)(45,63)(46,64)(47,61)(48,62) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48),(49,50,51,52),(53,54,55,56),(57,58,59,60),(61,62,63,64)], [(1,23,25,17),(2,24,26,18),(3,21,27,19),(4,22,28,20),(5,15,63,10),(6,16,64,11),(7,13,61,12),(8,14,62,9),(29,36,41,38),(30,33,42,39),(31,34,43,40),(32,35,44,37),(45,55,58,50),(46,56,59,51),(47,53,60,52),(48,54,57,49)], [(1,4),(2,3),(5,9),(6,12),(7,11),(8,10),(13,64),(14,63),(15,62),(16,61),(17,22),(18,21),(19,24),(20,23),(25,28),(26,27),(29,35),(30,34),(31,33),(32,36),(37,41),(38,44),(39,43),(40,42),(45,57),(46,60),(47,59),(48,58),(49,50),(51,52),(53,56),(54,55)], [(1,48,25,57),(2,58,26,45),(3,46,27,59),(4,60,28,47),(5,44,63,32),(6,29,64,41),(7,42,61,30),(8,31,62,43),(9,40,14,34),(10,35,15,37),(11,38,16,36),(12,33,13,39),(17,49,23,54),(18,55,24,50),(19,51,21,56),(20,53,22,52)], [(1,29),(2,30),(3,31),(4,32),(5,58),(6,59),(7,60),(8,57),(9,49),(10,50),(11,51),(12,52),(13,53),(14,54),(15,55),(16,56),(17,36),(18,33),(19,34),(20,35),(21,40),(22,37),(23,38),(24,39),(25,41),(26,42),(27,43),(28,44),(45,63),(46,64),(47,61),(48,62)]])
Matrix representation of C42.26C23 ►in GL8(𝔽17)
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 16 | 1 | 15 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 1 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 16 | 1 | 15 |
| 0 | 0 | 0 | 0 | 1 | 0 | 1 | 16 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 16 | 1 | 15 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 16 | 1 | 16 | 2 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 16 | 1 | 15 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
| 1 | 16 | 1 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 16 | 1 | 15 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 1 | 16 |
| 14 | 14 | 0 | 0 | 0 | 0 | 0 | 0 |
| 14 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 14 | 3 | 11 | 6 | 0 | 0 | 0 | 0 |
| 14 | 0 | 14 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 16 | 1 | 15 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(8,GF(17))| [0,1,16,16,0,0,0,0,0,16,0,1,0,0,0,0,1,1,0,0,0,0,0,0,0,15,0,1,0,0,0,0,0,0,0,0,0,16,1,1,0,0,0,0,1,0,16,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16],[0,16,1,1,0,0,0,0,1,0,16,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,16,1,1,0,0,0,0,0,1,0,0,0,0,0,0,1,16,0,1,0,0,0,0,0,2,0,16,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,1,0,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,15,16],[1,0,0,1,0,0,0,0,16,0,1,0,0,0,0,0,1,16,0,1,0,0,0,0,15,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,16,16,0,0,0,0,1,16,0,1,0,0,0,0,15,0,0,16],[14,14,14,14,0,0,0,0,14,3,3,0,0,0,0,0,0,0,11,14,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,15,0,1] >;
C42.26C23 in GAP, Magma, Sage, TeX
C_4^2._{26}C_2^3 % in TeX
G:=Group("C4^2.26C2^3"); // GroupNames label
G:=SmallGroup(128,1991);
// by ID
G=gap.SmallGroup(128,1991);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,219,675,1018,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=e^2=1,d^2=b^2,a*b=b*a,c*a*c=a^-1,d*a*d^-1=a*b^2,a*e=e*a,c*b*c=e*b*e=b^-1,b*d=d*b,d*c*d^-1=a^2*c,e*c*e=b*c,e*d*e=a^2*d>;
// generators/relations
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